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A survey on classical minimal surface theory. (English) Zbl 1262.53002

University Lecture Series 60. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-6912-3/pbk). 182 p. (2012).
Beside the topics discussed in the authors’ survey paper [Bull. Am. Math. Soc., New Ser. 48, No. 3, 325–407 (2011; Zbl 1232.53003)], in this monograph some new results are presented concerning properly embedded minimal planar domains in \(\mathbb R^3\) as well as many important recent advances in the theory of minimal surfaces. Among the topics considered, we may quote: the topological classification of minimal surfaces in \(\mathbb R^3\), the uniqueness of Scherk’s singly-periodic minimal surfaces, new aspects of the Calabi-Yau problem for minimal surfaces based on some recent work, a survey of Colding-Minicozzi theory for embedded minimal disks, the asymptotic behavior of minimal annular ends with infinite total curvature, the local removable singularity theorem for minimal laminations and applications.
After an introductory chapter containing the main definitions and background material (with short geometric and analytic descriptions of the important classical examples of properly embedded minimal surfaces in \(\mathbb R^3\)), a description of minimal surfaces with finite topology and more than one end is presented. Several methods for constructing properly embedded minimal surfaces of finite topology and for describing their moduli spaces are discussed.
Next, in Chapter 4, the authors present an overview of some results concerning the geometry, compactness and regularity of limits of locally simply connected sequences of minimal surfaces. In Chapter 5 they define and develop the notion of minimal lamination, viewed as a natural limit object for a locally simply connected sequence of embedded minimal surfaces. In Chapter 6, the “Ordering Theorem” for the linear ordering of the ends of a properly embedded minimal surface with more than one end is presented. Chapter 7 is devoted to conformal questions on minimal surfaces. The authors define and use the notion of universal superharmonic function for domains in \(\mathbb R^3\). The results are applied to explain the main steps in the proof of Theorem 1.0.1: A complete, embedded, simply connected minimal surface in \(\mathbb R^3\) is a plane or a helicoid.
A discussion and a more general classification result of complete embedded minimal annula ends with compact boundary and infinite total curvature in \(\mathbb R^3\) is found in Chapter 9. In Chapter 10, the authors complete their proof of Theorem 1.0.1. In Chapter 11, they examine some new global results in the classical theory and a deeper understanding of the local geometry of minimal surfaces obtained from the theoretical results presented in the previous chapters. Here are some topics discussed in this chapter: local picture theorem on the scale of curvature and topology, parking garage structure, fundamental singularity conjecture, local removable singularity theorem, quadratic curvature decay theorem.
In Chapter 12, the authors present some results on the geometry of complete embedded minimal surfaces of finite genus with a possibly infinite number of ends. They obtain also a classification of properly embedded minimal planar domains in \(\mathbb R^3\). Here the reader may find a proof of Theorem 1.0.2 concerning a finer classification of minimal surfaces in \(\mathbb R^3\). Next, he finds a topological classification of properly embedded minimal surface in \(\mathbb R^3\), the ordering theorem for the ends of a properly embedded minimal surface \(M\subset \mathbb R^3\) and some methods for constructing a smooth (nonminimal) surface in the isotopy class of a properly embedded minimal surface with prescribed topological invariants. In Chapter 14 the authors present some positive results concerning the Liouville conjecture for properly embedded minimal surfaces. In Chapter 15, they explain some classification results closely related to the procedure of minimal surgery. They discuss some aspects derived from the Calabi-Yau problems for complete minimal surfaces in \(\mathbb R^3\). The final chapter is devoted to a discussion of the main outstanding conjectures in the subject, which have lead to recent advances in classical minimal surface theory.

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q05 Minimal surfaces and optimization
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 1232.53003
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